Automatic Generation of Generating Functions for Chromatic Polynomials for Grid Graphs (and more general creatures) of Fixed (but arbitrary!) Width

By Shalosh B. Ekhad, Jocelyn Quaintance, and Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger)

Written: March 30, 2011.

Dédié à la mémoire de PHILIPPE FLAJOLET (1er décembre 1948 - 22 mars, 2011)

This short article is modest homage to the great guru who coined the terms analytic combinatorics, and singular combinatorics, and made us almost love analysis.

# Sample Input and Output for the Maple package KamaTzviot

• If you want to see explicit expressions (as rational functions of t and c) for the formal power series whose coefficient of tn in its Maclaurin expansion (with respect to t) would give you the chromatic polynoial of the m by n grid graph (the Cartesian product of a path of m vertices and a path of length n) for m=1 to m=6, the
the input gives the output.

• If you want to see explicit expressions (as rational functions of t and c) for the formal power series whose coefficient of tn in its Maclaurin expansion (with respect to t) would give you the chromatic polynoial of the m by n cylinder graph (the Cartesian product of a cycle of m vertices and a path of length n) for m=1 to m=6, the
the input gives the output.

• If you want to see the first 30 terms of the sequence of chromatic polynomials the m by n grid graph (the Cartesian product of a path of m vertices and a path of length n) for n=1 to n=30, and for m=1 to m=6, followed by the integer sequences for the number of colorings with 2,3,4, and 5 colors, the
the input gives the output.

• If you want to see the first 30 terms of the sequence of chromatic polynomials the m by n cylinder graph (the Cartesian product of a cycle of m vertices and a path of length n) for n=1 to n=30, and for m=1 to m=6, followed by the integer sequences for the number of colorings with 2,3,4, and 5 colors, the
the input gives the output.

• If you want to see the the chromatic polynomials for the n by n square grid graphs, for n between 1 and 7, the
the input gives the output.

• If you want to see the the generating function for the chrmoatic polynomials Mn(G,C) (see the article for the definition) where G is the 5-vertex graph (whose vertices are labeled {1,2,3,4,5})
G:={{1,2},{2,3},{3,4},{4,5},{1,3},{2,4},{3,5}}:
and C is the bipartite graph
C:={[1,1],[2,2],[3,3],[4,4],[5,5],[1,2],[2,1],[2,3],[3,2],[3,4],[4,3],[4,5],[5,4]}:
the input gives the output.

Added April 7, 2011: Here is an addendum written by Jocelyn Quaintance.
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger